Combination of Tensor Network States and Green's function Monte Carlo

Abstract

We propose an approach to study the ground state of quantum many-body systems in which Tensor Network States (TNS), specifically Projected Entangled Pair States (PEPS), and Green's function Monte Carlo (GFMC) are combined. PEPS, by design, encode the area law which governs the scaling of entanglement entropy in quantum systems with short range interactions but are hindered by the high computational complexity scaling with bond dimension (D). GFMC is a highly efficient method, but it usually suffers from the infamous negative sign problem which can be avoided by the fixed node approximation in which a guiding wave function is utilized to modify the sampling process. The trade-off for the absence of negative sign problem is the introduction of systematic error by guiding wave function. In this work, we combine these two methods, PEPS and GFMC, to take advantage of both of them. PEPS are very accurate variational wave functions, while at the same time, only contractions of single-layer tensor network are necessary in GFMC, which reduces the cost substantially. Moreover, energy obtained in GFMC is guaranteed to be variational and lower than the variational energy of the guiding PEPS wave function. Benchmark results of $J_1$-$J_2$ Heisenberg model on square lattice are provided.

arXiv comment4+ pages, 4 figures, close to published version

References in corpus (70)

• The density-matrix renormalization group in the age of matrix product states
• The density-matrix renormalization group
• Area laws for the entanglement entropy - a review
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Quantum Spin Liquid States
• Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations
• Spin Liquid Ground State of the $S=1/2$ Kagome Heisenberg Model
• Thermodynamic limit of the density matrix renormalization for the spin-1 Heisenberg chain
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• A class of quantum many-body states that can be efficiently simulated
• Classifying quantum phases using Matrix Product States and PEPS
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• Tensor renormalization group approach to 2D classical lattice models
• Projected wavefunction study of Spin-1/2 Heisenberg model on the Kagome lattice
• Stripe order in the underdoped region of the two-dimensional Hubbard model
• Classical simulation of quantum many-body systems with a tree tensor network
• Entanglement entropy of fermions in any dimension and the Widom conjecture
• Finite-Size Scaling of the Ground State Parameters of the Two-Dimensional Heisenberg Model
• Competing states in the t-J model: uniform d-wave state versus stripe state
• Accurate determination of tensor network state of quantum lattice models in two dimensions
• Corner Transfer Matrix Renormalization Group Method
• Long range Neel order in the triangular Heisenberg model
• Solutions of the Two Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms
• Coarse-graining renormalization by higher-order singular value decomposition
• Simulation of two dimensional quantum systems on an infinite lattice revisited: corner transfer matrix for tensor contraction
• A Constrained Path Monte Carlo Method for Fermion Ground States
• Studying Two Dimensional Systems With the Density Matrix Renormalization Group
• Gapless spin-liquid ground state in the $S=1/2$ kagome antiferromagnet
• Matrix product operator representations
• Tensor Network Renormalization
• Spin Liquid Ground State of the Spin-1/2 Square $J_1$-$J_2$ Heisenberg Model
• Gapless spin-liquid phase in the kagome spin-1/2 Heisenberg antiferromagnet
• Simulation of strongly correlated fermions in two spatial dimensions with fermionic Projected Entangled-Pair States
• The computational complexity of PEPS
• Absence of superconductivity in the pure two-dimensional Hubbard model
• Plaquette Ordered Phase and Quantum Phase Diagram in the Spin-1/2 J1-J2 Square Heisenberg Model
• The iPEPS algorithm, improved: fast full update and gauge fixing
• Plaquette valence bond solid in the frustrated Heisenberg quantum antiferromagnet on the square lattice
• Algorithms for finite Projected Entangled Pair States
• Direct evidence for a gapless $Z_2$ spin liquid by frustrating Néel antiferromagnetism
• Spontaneous plaquette dimerization in the $J_1-J_2$ Heisenberg model
• Renormalization of tensor-network states
• Critical level crossings and gapless spin liquid in the square-lattice spin-$1/2$ $J_1$-$J_2$ Heisenberg antiferromagnet
• Variational quantum Monte Carlo simulations with tensor-network states
• Loop optimization for tensor network renormalization
• Gapped spin liquid with ${\mathbb{Z}}_2$-topological order for kagome Heisenberg model
• Exploring frustrated spin-systems using Projected Entangled Pair States (PEPS)
• Loop updates for variational and projector quantum Monte Carlo simulations in the valence-bond basis
• Tensor renormalization of quantum many-body systems using projected entangled simplex states
• Strings, Projected Entangled Pair States, and variational Monte Carlo methods
• Quantum Spin Liquid in Spin 1/2 J1-J2 Heisenberg Model on Square Lattice: Many-Variable Variational Monte Carlo Study Combined with Quantum-Number Projections
• Simultaneous dimerization and SU(4) symmetry breaking of 4-color fermions on the square lattice
• Green Function Monte Carlo with Stochastic Reconfiguration: an effective remedy for the sign problem disease
• Proof for an upper bound in fixed-node Monte Carlo for lattice fermions
• Monte Carlo simulation with Tensor Network States
• Spatially inhomogeneous phase in the two-dimensional repulsive Hubbard model
• Ground-state phase diagram of the quantum $J_1-J_2$ model on the square lattice
• Gauge fixing, canonical forms and optimal truncations in tensor networks with closed loops
• Spin-1/2 J1-J2 Heisenberg antiferromagnet on a square lattice: a plaquette renormalized tensor network study
• Coupling quantum Monte Carlo and independent-particle calculations: self-consistent constraint for the sign problem based on density or density matrix
• Gradient optimization of finite projected entangled pair states
• Optimized contraction scheme for tensor-network states
• Incorporation of Density Matrix Wavefunctions in Monte Carlo Simulations: Application to the Frustrated Heisenberg Model
• Variational Monte Carlo method for fermionic models combined with tensor networks and applications to the hole-doped two-dimensional Hubbard model
• Projector quantum Monte Carlo with matrix product states
• Very High Precision Determination of Low-Energy Parameters: The 2-d Heisenberg Quantum Antiferromagnet as a Test Case
• Variational Monte Carlo simulations using tensor-product projected states
• A generalized Lanczos method for systematic optimization of tensor network states

Cited by in corpus (5)

• Augmenting Density Matrix Renormalization Group with Disentanglers
• Unbiasing Fermionic Auxiliary-Field Quantum Monte Carlo with Matrix Product State Trial Wavefunctions
• The Green's function Monte Carlo combined with projected entangled pair state approach to the frustrated $J_1$-$J_2$ Heisenberg model
• All-mode Renormalization for Tensor Network with Stochastic Noise
• A hybrid method integrating Green's function Monte Carlo and projected entangled pair states